Randomized Congruence Closure

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Randomized Congruence Closure

• CS263 Project Presentation
• Ambuj Tewari

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Outline

• Theory of equality with uninterpreted functions
  and linear arithmetic
• Shostaks algorithm
• Randomized approach
• Running Time Comparisons

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Theory of Equality

• A theory consists of
• Function and predicate symbols
• A set of inference rules
• For example,
• f, g, h, (uninterpreted functions symbols)
• Rules (T ranges over sets of equalities)

(congruence)
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Theory of Equality with Arithmetic

• Example,
• To add linear arithmetic, we need
• Canonizer
• Solver
• More rules

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Theory of Equality with Arithmetic

• Example,
• Note that there is no multiplication except with
  constants (hence linear)
• We need a decision procedure to decide statements
  of the form

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Shostaks Algorithm

• Provides a decision procedure for theory of
  equality combined with canonizable and solvable
  theories
• Uses canonizing function s and solver solve as
  black-boxes
• Applicability is not limited to linear arithmetic
• Shostaks algorithm is essentially a clever
  combination of congruence closure and gaussian
  elimination with back substitution
• Discovered in 84
• Cyrluk et. al. 96 fixed some bugs
• Ruess and Shankar 01 claim to give formal
  proofs of correctness (soundcomplete) and
  termination

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Randomized Approach

• A set of equalities T, describes a subspace S(T)
  in an n-dimensional world (n number of
  variables)
• Key idea is to represent a subspace by a set of
  random points in that subspace
• To test if T ab, test whether the random
  points in S(T) satisfy ab (small probability of
  error)

• A line is parametrized by one variable
• A plane that doesnt contain it will have at
  most one intersection with it
• Probability that all d points happen to be the
  bad point (1/d)t

bad point
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Randomized Approach

• Gulwani and Necula 02 define an operation
  adjust(S,e)
• S contains a certain number, say m, of points
  drawn randomly from S(T)
• adjust returns m-1 points in S(T e0)
• If there were no uninterpreted functions then one
  adjust would suffice while processing a new
  equation
• After processing an equation if we find that, say
  xy, then we have to adjust for f(x)f(y) also
• Since we lose one point after each adjust, the
  probability of error is less than
  where r
• of random points, a of adjusts

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Running Time Comparisons

• Shostak implementation was mine not very
  optimized (now have access to a
  commercial-quality implementation)
• Randomized implementation (provided by Sumit
  Gulwani) highly optimized random numbers
  precomputed
• Both implementations computed over the field
  Z4093
• Time spent in parsing input not included
• Couldnt find commercial benchmarks
• Test cases were randomly generated by a program
  may not reflect what is encountered in real world

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No uninterpreted functions
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With uninterpreted functions

• 10 variables
• 5 uninterpreted functions with arity up to 3
• 20 equations
• Up to 2 levels of nesting of function symbols